Completely regular space
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
In the T family (properties of topological spaces related to separation axioms), this is called: T3.5
This article is about a basic definition in topology.
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Definition
A topological space is termed completely regular if it satisfies the following equivalent conditions:
- Given any point and any closed subset, there is a continuous function on the topological space that takes the value at the point and at the closed subset
- It occurs as the underlying topological space of a uniform space
- It possesses a compactification
Formalisms
In terms of the subspace operator
This property is obtained by applying the subspace operator to the property: compact Hausdorff space
Relation with other properties
Stronger properties
- Normal space
- Underlying space of a topological group
- Metrizable space
Metaproperties
Hereditariness
This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces
Any subspace of a completely regular space is completely regular.
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
An arbitrary product of completely regular spaces is completely regular.